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Zimmerman run11.tex V1 - January 5, 2008 1:08pm Page 248

11. The Structure of Gunk: Adventures

in the Ontology of Space

Jeffrey T. Russell

1. points and gunk

Here are two ways space might be (not the only two): (1) Space

is pointy. Every finite region has infinitely many infinitesimal,

indivisible parts, called points. Points are zero-dimensional atomsof space. In addition to points, there are other kinds of thin bound-

ary regions, like surfaces of spheres. Some regions include their

boundariesthe closed regionsothers exclude themthe open

regionsand others include some bits of boundary and exclude

others. Moreover, space includes unextended regions whose size

is zero. (2) Space is gunky.1 Every region contains still smallerFN:1

regions there are no spatial atoms. Every region is thick there

are no boundary regions. Every region is extended.

Pointy theories of space and space-time such as Euclidean space

or Minkowski spaceare the kind that figure in modern physics.

A rival tradition, most famously associated in the last century with

A. N. Whitehead, instead embraces gunk.2 On the WhiteheadianFN:2view, points, curves, and surfaces are not parts of space, but rather

abstractions from the true regions.

Three different motivations push philosophers toward gunky

space. Thefirst is that thephysical space (space-time)of ouruniverse

might be gunky. In Quinean terms, spatial regions are posits,

invoked to explain what goes on with physical objects; the main

reason to believe in point-sized regions is the role they play in our

Dean Zimmerman has been a relentless source of suggestions and criticismthroughout the writing of this chapter. Thanks are also due to Frank Arntzenius,Kenny Easwaran, Peter Forrest, Hilary Greaves, and John Hawthorne for helpful

comments.1 David Lewis coined the term gunk to describe infinitely divisible atomlessobjects (1991).

2 For historical background, see Zimmerman 1996b.


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The Structure of Gunk 249

physical theories. So far it looks like points are doing well in that

regard despite their uncanniness: all of our most successful theories

representspace-time as a manifold of points. But do the points really

do important theoretical work, or are they mere formal artefacts,

scaffolding to be cast off of our final theory? Modern physics does

offer some evidence for the latter. Frank Arntzenius observes that

the basic objects of quantum mechanics are not fields in pointy

space, but instead equivalence classes of fields up to measure zero

differences: point-sized differences between fields are washed out

of the theory (2007). This is suggestive: perhaps the points dont

belong in the theory in the first place.

A second motivation is interest in possibility rather than actuality.

For some metaphysicians, whether or not our own universes spaceis pointy or gunky is of only incidental interest; the real question

is what ways space could bein the sense of metaphysical rather

than epistemic possibility. This question is pressing due to the

role of material gunk in recent debates. Many philosophers have

the intuition that atomless material objects are genuinely possible,

even if atomism happens to be true in the actual world, and certain

views have beencriticized for failing to accommodatethis intuition.3FN:3

Furthermore, there are reasons to think that gunky space is the right

environment for material gunk. One road to that conclusion is by

way of harmony principles that link the composition of material

objects to the composition of the spatial regions they occupy.4FN:4

Another route to gunky space is by way of contact puzzles.5

FN:5

Suppose two blocks are in perfect contact. If space is pointy, then

there is a region of space at the boundary between the two blocks.

Either this region is empty which apparently contradicts the claim

that the two blocks are touchingor else it is occupied by both

blocks which looks like disturbing co-location or else some

strange asymmetry is at work. Perfect contact may not take place

in the actual world, but to the extent that one thinks it a genuine

(metaphysical) possibility, this should incline one toward gunk. (Its

plausible, for instance, that gunky space is the right environment

for corpuscular Newtonian mechanics, where collisions are taken

seriously.)

3 E.g. in Sider 1993.4 Cf. Gabriel Uzquiano, Mereological Harmony (unpublished draft).5 Cf. Zimmerman 1996a.


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250 Jeffrey T. Russell

A third reason to investigate gunky space is to formalize the

psychology of space. Regardless of whether phenomena like perfect

contact are physically or even metaphysically possible, they surely

play into common sense spatial reasoning. Rigorously capturing

this kind of reasoning is important for formal semantics, cognitive

science, and artificial intelligence (Cohn and Varzi 2003).

These different motivations all call for a clear account of space

without points. My interest is in framing theories of gunky space

that are as formally adequate and precise as the theories of pointy

space on offeran ambition many share. One pioneer is Alfred

Tarski, who presents an elegant formalization of Whiteheadian

geometry, which he describes as a system of geometry destitute

of such geometrical figures as points, lines, and surfaces, andadmitting as figures only solidsthe intuitive correlates of open

(or closed) regular sets of three-dimensional Euclidean geometry

(1956). (Note, though, that the issues in this chapter are independent

of spaces particular dimension or curvature.)

In addition to framing precise theories, it is useful to construct

models of gunky space. The open regular sets are a model for

Tarskis geometry, since they satisfy his axioms.6 This demonstratesFN:6

that his theory is logically consistentas long as the Euclidean

theory, in whose terms the model is couched, is consistent. (Of

course, the model is not meant to show what space really is. If

gunky space were really an abstraction from pointy space, then the

gunky space would only exist if the pointy space did as wellthiswould make things awfully crowded.)

Recently, though, Whiteheadian space has come under fire:

Peter Forrest (1996) and Frank Arntzenius (2007) have indepen-

dently offered parallel arguments that challenge the coherence

of some gunky accounts.7 Theories like Tarskis run into seri-FN:7

ous trouble when we ask questions about the precise sizes of

regions. In this chapter I present a generalization of the For-

rest Arntzenius argument that makes clear which features of

6 An open regular set is an open set which is the interior of its closure. The interiorof x (int x) is the largest open set contained in x (equivalently, the set of xs interior

points). The closure of x (cl x) is the smallest closed set that contains x (equivalently,the set of xs limit points). For example: the open interval (0, 1) is an open regular set;the union of open intervals (0,1) (1,2) is not.

7 John Hawthorne and Brian Weatherson present a related argument (2004).


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space are incompatible. The objection thus sharpened, I take up

gunks defense, considering three responses: Arntzeniuss, For-

rests, and a new response. The Forrest Arntzenius argument

requires some concessions, but it does not force us to give up

gunky space.

2. three kinds of spatial structure

Before presenting the main argument in Section 3, I need to

lay some formal groundwork for the relevant spatial structures.

Along the way I isolate three different constraints on Whiteheadian

space.

2.1 Mereology

The first of these structures is mereology: a region x may be part of

a region y (for short, x y); synonymously: y contains x, or x is a

subregion of y (in the weak sense where every region is a subregion

of itself).8 My main argument only requires two properties of thisFN:8

relation (x, y, and z are arbitrary regions throughout):

(1) x x. (Reflexivity)

(2) If x y and y z, then x z. (Transitivity)

These are very weak conditions: any two-place relation at all has a

transitive and reflexive extension.Some results besides the main argument depend on slightly more

structure. Adding one further proposition yields the axiom system

of Ground Mereology (M):

(3) If x y and y x, then x = y. (Antisymmetry)

Throughout this chapter I invoke the three axioms of M without

apology. I dont make use of any mereological principles beyond

these without due warning.

I also use the following definitions:

(D1) x and y overlap iff there is a region z such that z x and

z y. Regions that do not overlap are disjoint.

(D2) x is a proper part of y (x < y) iff x y and x = y.

8 The definitions and axioms in this section are adapted from Varzi 2006.


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Intuitively, a proper part must have something left oversome-

thing that supplements it. This intuition has formalizations of

various strengths:

(4) If x < y, then y has a part that is disjoint from x. (Weak

Supplementation)

(5) Unless x y, y has a part that is disjoint from x. (Strong

Supplementation)

An even stronger formulation is that x must have some remainder in

ysome part which exactly makes up the difference between them.

(D3) A region z is the remainder of x iny (or the mereological

difference ofy and x, denotedy x) iff: for every region

w, w z iff w is a part of y that is disjoint from x.

A remainder is a maximal supplement. This gives rise to the third

and strongest supplementation principle:

(6) Unless x y, x has a remainder iny. (Remainder Closure)

Besides taking differences between regions, we can also take sums

of regions.

(D4) A region f is a fusion (or mereological sum) of a set of

regions S iff: for every region x, x overlapsf iff x overlaps

some y S.

Intuitively, the fusion of the ys is the region that includes all of the

ys and nothing more. If Strong Supplementation holds, then any

set of regions has at most one fusion. As we have various supple-mentation principles, we also have various composition principles.

This is one standard principle:

(7) For any x andy, someregion isa fusionof x andy (denoted

x y). (Sum Closure)

The principles listed so far are together equivalent to the axioms of

Closure Extensional Mereology (CEM). A much stronger composition

principle is the following:

(8) Any set of regions has a fusion. (Unrestricted Composi-

tion)

Under this principle, mereological sums exist for arbitrary finite

or infinite collections of regions. Adding Unrestricted Compositionto CEM yields the system of General Extensional Mereology (GEM),

also called simply standard mereology. Euclidean and Tarskian


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geometry both satisfy GEM, but (to reiterate) I dont assume

standard mereology in what follows.

Points have a distinctive mereological feature: they are atoms,

regions with no proper parts. Pointless space has a corresponding

feature: it is atomless.

(9) Every region has a proper part. (Mereological Gunk)

2.2 Topology

Topology describes general shape properties such as connected-

ness, continuity, and having a boundary. Mathematical orthodoxy

casts topological structure in terms of properties of point-sets:

the open sets are taken as primitive, and other structures aredefined in terms of them. This is of little use if there are no

points. Instead Ill follow the tradition of Whitehead, Bowman

Clarke (1981), and Peter Rper (1997), which instead character-

izes topology in terms of a primitive relation of connectedness.

Intuitively, two regions x and y are connected (x y) if they are

adjacent, including if they overlap. The corresponding notion in

point-set topology is sharing a limit point; i.e. having overlapping

closures.9FN:9With connectedness on hand, we can define a number of other

important notions.10FN:10

(D5) x is a boundary of y iff every part of x is connected to y

and also to some region disjoint from y.A boundary region is tightly sandwiched between some region and

its complement.

(D6) x is open iff no part of x is a boundary of x.

9 Tarski does not take connection as primitive in his geometry. Instead, Tarskitreats sphericality as a primitive property of regions. The spheres impose a metricstructure on the space, which induces a topology. Tarski (with some ingenuity)frames a purely mereological condition for two spheres to be concentric. Then wecan define x and y are connected as There exists some sphere s such that everysphere concentric with s overlaps both x and y. This parallels sharing a limit point inpoint-set topology: the set of concentric spheres is an ersatz point.

10 The following connection axioms suffice to prove the claims in this section:

(i) x x. (Reflexivity)(ii) If x y, then y x. (Symmetry)(iii) If x y and y z, then x z. (Monotonicity)


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(D7) A region x is an interior part of y iff x y and x is not

connected to any region disjoint from y.11FN:11

It follows from the definitions that for any x y, either x is a

boundary of y, or else x contains an interior part of y. In particular,

every open region has an interior part. Moreover, except when a

region is disconnected from the rest of space, every interior part is

a proper part.

Whiteheadian space isnt just mereologically distinctive, but also

topologically distinctive: there are no boundary regions. We can

formulate this as a second condition on gunk:

(10) No region is a boundary. (No Boundaries)

Or equivalently,

(11) Every region is open.

This entails, and in the presence of Strong Supplementation is

equivalent to,

(12) Every region has an interior part.12FN:12

I call space that satisfies No Boundaries topologically gunky. Topo-

logical Gunk is a natural extension of Mereological Gunk: not only

does every region have a proper part, it has a part which is strictly

inside of it.

I draw on one further topological notion. A topological basis is

a collection of open regions with which all of the open regions

can be characterized: any open region is a mereological sum of

basis regions. An important topological basis in Euclidean space

is the set of rational spheres: the open spheres with rational center

coordinates and rational radii. This basis is countable. Any space

that is shaped like Euclidean space, whether pointy or gunky, has a

countable basis. For instance, the regions in Tarskis geometry that

correspond to the rational spheres make up a countable basis there.

Since in topologically gunky space every region is open,

11 When Strong Supplementation holds, the clause x y is redundant.12 (11) entails (10) because if x is a boundary of an open region y, then x is disjoint

from y and connected to y at every part, hence not open. (12) entails (10) by thefollowing argument. Suppose x is a boundary of y. First case: x y. By Strong Sup-

plementation, some z x is disjoint from y. But every part of xand so every part ofzis connectedtoy, so z hasno interior part. Second case:x y. Every z x is connect-edtosome w disjoint fromy, andthusdisjoint from x aswellso x hasnointeriorpart.


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(13) If space is topologically gunky and B is a topological

basis, every region contains an element of B.

This is important.

2.3 Measure

Besides mereology and topology, regions have sizes. This feature

is formalized using a measure function, m, which assigns sizes (rep-

resented by non-negative real numbers up to infinity) to regions

of space (Munroe 1953). A measure function should satisfy this

condition:

(14) If x and y are disjoint and f= x y, then m(f) = m(x)+

m(y). (Finite Additivity)FiniteAdditivity has an important generalization, which is an axiom

of standard measure theory:

(15) For any countable set of pairwise-disjoint regions S, if

f is a fusion of S then m(f) =

xS m(x). (Countable

Additivity)

With Remainder Closure, this has an important consequence:

(16) For any countable set of regions S, if f is a fusion of S

then m(f)

xS m(x). (Countable Subadditivity)

The whole is no larger than the sum of its parts. I discuss this

condition in Section 4.2.

Now we turn to measuring gunk. In pointy space, there are unex-tended regions regions (like points, curves, and surfaces) that have

measure zero. Gunky space is supposed to be free of unextended

regions:13FN:13

(17) Every region has positive measure. (No Zero)

Gunky space also shouldnt have discrete chunks, regions with

no parts smaller than some finite size. Rejecting chunks means

committing to the following principle:

(18) Every region has an arbitrarily small part: i.e., for every

region x, for any > 0 there is some y x such that

m(y) . (Small Regions)

13 Arntzenius and Hawthorne propose this condition (2005).


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Small Regions holds automatically in pointy space: every region

contains a point with size zero. It also follows from Mereological

Gunk, given Remainder Closure and Finite Additivity.14 The con-FN:14

junction of No Zero and Small Regions entails (and again given

Remainder Closure and Finite Additivity, it is equivalent to):

(19) Every region has a strictly smaller part. (Measure-Theor-

etic Gunk)

As having an interior part is a natural topological elaboration on

having a proper part, having a smaller part is a natural measure-

theoretic elaboration. Note that since each of the three ways of

being gunky constrains a different sort of structure, they need not

all stand or fall together.

3. disaster

In fact, these structural constraints cant allhold.The followingargu-

ment by Arntzenius (in this volume), gives a picture of why not.

(Forrest (1996) offers a parallel argument using a different construc-

tion.) For simplicity, consider one-dimensional gunky spacea

lineand let the Big Interval be a one-inch interval in this space.

Let a be the 14 -inch segment in the middle of the Big Interval. Then

consider the two subregions of the Big Interval flanking a, and let b1and b2 be intervals

12 of

18 inch long (i.e.

116 inch) in their respective

middles. This divides r into a, b1, b2, and four equal regions sur-rounding them. In each of the latter four regions, pick out a region

that is 14

of 116

inch long (i.e. 164

inch); call these c1 through c4. And

so on. Call a, b1, b2, . . . the Cantor Regions.

The Cantor Regions include one 14 -inch interval, two whose

lengths add up to 18 inch, four whose lengths add up to1

16 inch, etc.

So the sum of their lengths is 14 +18 +

116 + . . . =

12 inch strictly less

than the length of the Big Interval. And yet: if space is topologically

14 Except in the degenerate case where some infinite region only has infinitesubregions. Proof. Consider an arbitrary region x with positive finite measure. ByMereological Gunk, x has a proper part y, and by Remainder Closure (its easy toshow) x = y (xy), so by Finite Additivity m(x) = m(y)+ m(xy). It follows that

at least one of m(y) and m(xy) must be less than or equal to 12 m(x). Repeating thisn times procures a part of x no larger than m(x)/2n, which is eventually smaller thanany positive .


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gunky, then the Big Interval is the fusion of the Cantor Regions (Ill

defer the details to the main argument below). We have a set of

regions whose lengths add up to half an inch, but whose fusion is a

whole inch long: this is serious trouble.

Before responding to this predicament, I want to sharpen the

troubling feeling with my own more general argument.15FN:15

Theorem. The following five theses are inconsistent:

1. Space has a transitive and reflexive parthood relation.

2. Space has a topology with a countable basis.

3. Space is topologically gunky.

4. Space has a non-trivial countably subadditive measure.16FN:16

5. Every region has an arbitrarily small subregion.

Proof. Consider a region with positive finite measure the BigRegionand call itsmeasureM.Letthe Insidersbethe basis elements

that are parts of the Big Region. Since the basis is countable, there

are countably many Insiders, so we can enumerate them. For

i = 1,2, l . . ., pick a subregion of the ith Insider whose measure is

less than M/2i+1; call these subregions the Small Regions. The Small

Regions sizes add up to strictly less than M.

Even so, the Big Region is a fusion of the set of Small Regions. To

show this, we need to show that every region that overlaps a Small

Region overlaps the Big Region, and vice versa. The first direction

is obvious: each Small Region is part of the Big Region. Now let x

be any region that overlaps the Big Region, so x and the Big Region

have a part in common, y. By Topological Gunk y contains a basiselement b, which must be one of the Insiders, and so b contains a

Small Region. Since b x, x overlaps a Small Region.

Therefore the Big Region fuses the Small Regions, as advertised.

But the Small Regions are too small for that! This means that Count-

able Subadditivity fails.

Standard mereology, topology, and measure theory, even when

stripped to a bare skeleton, are inconsistent with topological gunk.

15 My argument is more closely analogous to Forrests variant than ArntzeniussForrests construction, though, depends specifically on pointy representations ofregions: he appeals to regions that are well-represented by neighborhoods of points Q1

with rational coordinates. Moreover, both Arntzeniuss and Forrests constructionsdepend on assumptions about the geometry of space. By contrast, my construction ispurely topological.

16 By non-trivial I mean that at least one region has positive finite measure.


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Any account of gunk that commits to all five theses is incoherent

so things look bad for Whiteheadian space.

3.1 Aside: measurability

My argument officially requires a measure function that is defined

for every region of space. This may seem too strict a requirement;

the Vitali theorem and the more notorious BanachTarski theorem

show that in Euclidean space there are sets of points that cannot be

consistently assigned a measure (these theorems require the Axiom

of Choice). Thus the standard measure function in Euclidean space

(Lebesgue measure) is defined only on a distinguished collection

of sets, which are called measurable. Here are two metaphysical

interpretations of the mathematics: one might say that some regions

in Euclidean space have no sizes; or one might say that the true

regions in Euclidean space correspond to just the measurable sets.17FN:17

(On the latter interpretation, Unrestricted Composition fails: an

arbitrary set of regions does not necessarily compose a regionfor

there are sets of point-sized regions which do not. Fusions of

arbitrary countable sets of regions still exist, but all bets are off for

the uncountable sets.)

We hope that gunky space will free us from such discomforts.

Arntzenius and Hawthorne propose a requirement on gunk to this

effect (they call it Definition) (2005):

(20) Every region is measurable. (Measurability)Note that my argument requires no composition principles, so

Measurability can be had cheaply by banishing any supposed

exceptions, as in the latter response to BanachTarski. But even

admitting non-measurable regions gives no relief from my argu-

ment. Small Regions guarantees the existence of regions with small

measures, which are a fortiori measurable, and this supplies enough

measurable regions to make the argument work. Even if we revise

this thesis to say only that every measurable region has an arbi-

trarily small part (so we dont get measurable regions for free)

the argument still succeeds with an unobjectionable extra premise:

every basis element (e.g. every sphere) is measurable. This permits

restricting the measure function far beyond what would suffice to

17 Arntzenius discusses these (2007).


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accommodate BanachTarski, without escaping an inch from my

argument. Measurability is not the culprit.

4. escape

The generalized ForrestArntzenius argument doesnt leave very

many places for the gunk theorist to turn. One way out is to reject

one of the three kinds of structure: in fact space does not have,

or (for the theorist who only wants the possibility of gunk) does

not necessarily have measure, topology, or mereology. Deleting

any of these is dizzying; perhaps sizeless or shapeless spaces are

broadly possible even so, but henceforth I assume that our gunk

theorist wants to hold onto at least the rudiments of parthood, size,and shape. This theorist has to deny one of the five inconsistent

conditions on these structures.

The mereological conditionthat parthood is reflexive and tran-

sitive is so weak as to be beyond reproach.

What about givingup a countabletopology? There are topological

spaces that do not have countablebases,but generallyspeaking they

are exotic infinite dimensional affairs. Such space would be shaped

nothing like Euclidean space or any other ordinary manifold. There

may still be interesting possible spaces to be explored in this

direction, but for now I leave that exploration to others.

And Small Regions? We could imagine chunky space with some

smallest volumesay on the Planck scalewhich is a floor onthe size of regions. But for space like this to be gunky it would

have to be non-standardfor as I observed in Section 2.3, denying

Small Regions while holding Mereological Gunk entails that either

Remainder Closure or Finite Additivity fails. In what follows I

consider some revisions to standard mereology and measure theory

in their own right, but none of themgainappealfor the gunk theorist

by being conjoined with the denial of Small Regions. So I pass over

this route, too.

This leaves two claims open to scrutiny: Topological Gunk and

Countable Subadditivity. Rejecting either of these bears a serious

cost, but does present live options for the gunk theorist. Arntze-

niuss response denies Topological Gunk, while Forrests response(in a roundabout way involving non-standard mereology) denies

Countable Subadditivity. I will discuss each of these avenues, as


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well as a third response that appeals more directly to non-standard

measure theory.

4.1 Gunk with boundaries

Arntzenius (following Roman Sikorski and Brian Skyrms) propos-

es a conception of gunky space that departs rather dramatically

from Whiteheads (Arntzenius 2007; Sikorski 1964; Skyrms 1993).

Arntzeniuss space obeys standard mereology, it has a topology

with a countablebasis, it has a countably additive measure function,

and moreover, it is both mereologically and measure-theoretically

gunky. It is not, however, topologically gunky, and thus escapes

the problem at hand.To motivate Arntzeniuss construction of a model for this kind

of space, lets revisit Tarskis model the open regular sets from

another angle. We start with orthodox pointy Euclidean space and

then smudge it, leaving out some distinctions the pointy theory

makes but retaining the most important structure. There is a gen-

eral method for smudging certain unimportant elements out of a

mereology18 Arntzeniuss metaphor for this is putting on blurryFN:18

glasses. The method works as long as the unimportant elements

form an ideal: any part of an unimportant element must be unim-

portant, and the fusion of any two unimportant elements must be

unimportant. To model Tarskis space,we smudgethe boundaries of

regions; its not hard to check that the boundary regions are an ideal.Once we have picked an ideal of unimportant elements, we define

an equivalence relation on the original space: two regions x and

y are considered equivalent just in case xy and y x are both

unimportant.19 (It is easy to show that for any ideal this is indeedFN:19

an equivalence relation.) This presents us with a natural collection

18 Assuming CEM, i.e. that the mereology forms a Boolean algebra. The result iscalled a quotient algebra (Sikorski 1964: 29 ff.).

Note: a Boolean algebra conventionally includes bottom and top elements: anull region contained by everything and a universal region containing everything.There aretrivial translationsbetween mereologywith a null regionand without it:forinstance, in the presence of a null region, disjoint regions have exactly one common

part. Introducing a null region is a technical convenience on par with introducing as a value for functionsthat increase without bound, or # to indicate truth-valuegaps.

19 Including when xy (or y x) is null i.e., when x < y (or y < x).


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of smudged objects: the equivalence classes under this relation

represent regions of gunky space.20 In the case of Tarskian space,FN:20

this amounts to considering two Euclidean regions equivalent if

they agree up to their boundary points.

The representatives inherit mereology from the original space: if

gunky regions r and s are represented by equivalence classes X and

Y, then r s iff for every x X and y Y, xy is unimportant.

They also inherit topological structure from the Euclidean topology:

r s iff for every x X and y Y, x y. Showing that these satisfy

the standard axioms is straightforward.21 (Choosing a measureFN:21

for Tarskian space is more involved; I discuss this in Section 4.4.)

Each Tarskian equivalence class contains a unique open regular set,

which we can treat as a regions canonical representative; in fact,the model we have just constructed is isomorphic to Tarskis.

This model gives primary importance to the topological gunk

condition: we use blurry glasses that cant see boundaries, ensuring

that the No Boundaries condition will be met. As we have seen,

this condition is problematic. Arntzenius responds by instead priv-

ileging the measure-theoretic features of gunk, in particular No

Zero. His model is constructed using blurry glasses that are blind

to Euclidean regions with (Lebesgue) measure zero. These regions

also form an ideal, so we can follow exactly the same construction:

gunky regions are represented by equivalence classes of regions that

have measure zero differences. We obtain mereology and topology

as in the preceding paragraph,22

and we can set the size of a regionFN:22

20 One of these equivalence classesthe ideal itselfrepresents the null region.This class can simply be discarded from the final structure.

21 More carefully: in general this process results in a Boolean algebra; whetherthe algebra is completei.e., whether Unrestricted Composition holds needs to bechecked in individual cases. In the instances I consider, it does hold.

Its not clear what the standard axioms are for connection, but at any rate thedefined relation is reflexive, symmetric, and monotonic (Footnote 9), as well asobeying Rpers decomposition principle,

(iv) If x (y z) then x y or x z.

Finding a full adequate set of topological axioms in terms of connection remainsa major undertaking. Rper offers a set of ten axioms, but his tenth axiom isequivalent to the problematic Topological Gunk condition (and thus is not satisfied

by Arntzeniuss space); removing this axiom undoes his method of recovering pointsfrom pointless topology.

22 Arntzenius uses a different definition of connection. Say p is a robust limitpoint of x iff for every open neighborhood u p, m(u x) > 0. (Measure-equivalent


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to be the (Lebesgue) measure of any member of its representative

equivalence class, since equivalent regions have the same measure.

This yields a model of measure-theoretically gunky space couched

in the terms of standard pointy space. It follows that the consistency

of the one reduces to the consistency of the other: the prospects look

bright for Arntzenian space.

On the other hand, this space has some surprises for the gunk the-

orist. Consider again the Cantor Regions from Section 3. Arntzenian

space harbors both the Big Interval and a proper part of it that fuses

the Cantor Regionsthe Cantor Fusion. Moreover, it has a third

region making up the difference between themthe Cantor Dust.

This is a peculiar region. While it is extended (its size is positive),

it has no interior, and it is a boundary of the Cantor Fusion: trueto form, Arntzenian space violates the topological gunk condition.

Furthermore, the Cantor Dust is entirely scattered: every part is

divisible into disconnected parts. It is tempting to describe the

region as a fusion of uncountably many scattered pointsbut of

course this is the wrong description, since there are no points.

In Euclidean space, one can completely characterize a region in

terms of the points that make itup: the points are a mereological basis.

Likewise in Tarskian space, the rational spheres are a mereological

basis. Arntzenian space has no analog to either the points or

the spheres in this respect. The points are an atomic basis for

Euclidean space; the spheres are a countable basis for Tarskian

space. If Arntzenian space has any set of fundamental parts itmust be neither atomic nor countable. This isnt a crippling deficit,

but it helps explain why regions like the Cantor Dust are difficult

to get an intuitive grip on.

Arntzenian space is gunky, but in a weaker sense than full-

blooded Whiteheadian space. Space could be gunky in a weaker

sets have the same robust limit points; we can treat the robust closed sets ascanonical representatives.) Arntzeniuss condition for the connectedness of regionsrepresented by X and Y is that some p is a robust limit point of every x X and

y Y. This is equivalent to my condition. (Say x and y have no robust limit points incommon. Then every point in clx cly has a neighborhood u such that m(u x) = 0or m(u y) = 0. Since the Euclidean topology has a countable basis, there are

countable collections of open sets {vi} and {wi} such that clx cly

vi

wi andfor each i, m(vi x) = m(wi y) = 0. By countable additivity, x

vi and y

wi

are measure-equivalent to x andy (respectively) and theyhave disjoint closures thatis, they are not connected. The converse is clear.)


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sense still, satisfying Mereological Gunk but neither the topological

nor the measure condition. The now-familiar smudging method

produces a model for such space: we look at Euclidean space

through blurry glasses blind to regions made of finitely many points,

and define mereology, topology, and measure in parallel fashion to

Arntzeniuss space. The resulting space underscores how separate

the different gunk conditions are from one another: this space is

very much like ordinary Euclidean space in that it has bound-

aries, unextended regions, zero-dimensional regions, distinctions

between open and closed spheresbut even so, it has no atomic

regionsno points. For those whose interest in gunk springs from

purely mereological concerns, this bare-bones gunk should suffice.

4.2 Interlude: foundations of measure

One way to save gunk is to give up the topological condition and

settle for a weaker kind of gunkiness. But some gunk theorists for

instance, those who turn to gunk to solve contact puzzlesmay be

less willing to admit violations of No Boundaries. For these theorists,

the main alternative is to deny Countable Subadditivity. This is

a natural response for those in the WhiteheadTarski tradition:

the Forrest Arntzenius problem arises because Whitehead and

Tarski simply didnt take measure into account in their theories.

In particular, Tarskis geometry of solids satisfies four of the

contradictory five theses, but breathes not a word of measure.Since in standard mereology Countable Subadditivity follows

from Countable Additivity, I turn to the latter principle. Is there

a plausible theory of full-fledged measurable space without this

condition? Answering this question requires a detour into the

philosophical foundations of measure theory. So far I have treat-

ed sizes as if they were fundamental numerical tags on things.

This is obviously a fiction; the numbers and arithmetic operations

on them arise from other features of the world. Treating sizes

as numbers is convenient, but can mislead: it gives the illusion

that numerical operations are automatically meaningful, obscuring

underlying assumptions.23 We must make sense of sums of sizes,FN:23

and specifically infinite sums.

23 Compare Ernest Nagels comments (1931: 313).


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Say we have some determinable property of regions. For this

property to be size, it has to have quantitative structure. First, the

determinates should at least be ordered: it must be sensible to say

that this objects size is greater or less or the same as that objects

size. This guarantees only a fairlyweak kind of quantitativeness in

the medieval jargon, that of intensive qualities. Size, though, is an

extensive quality: it has algebraic structure such that sizes can beadded. This operation isnt literally numerical addition, but its

structurally analogous.

The algebra of sizes does not float free: it must correspond to a

natural relation on the objects that have sizes; Ill call this relation

aggregation.24 Aggregation imposes an algebraic structure on thingsFN:24

that have sizes; this structure is what underpins addition of sizes.The sum of two things sizes is the size of their aggregate.

(21) If x is the aggregate of y and z, then the size of x is the

sum of the size of y and the size of z. (Finite Additivity)

This additivity condition, I claim, has the status of a conceptual

truth: without it there is no intelligible talk of adding sizes at all.

What is aggregation? Its natural to say its mereological structure:

x is the aggregate of y and z just in case y and z are disjoint regions

whose mereological sum is x. Then the Finite Additivity condition

above amounts to the condition by that name in section 2.3: the sum

of two sizes is the size of the mereological sum of disjoint regions

that have those sizes. Composition is what gives additive structure

to sizes. For now, though, Ill go on saying aggregation withoutcommitting to which relation it is.

This accounts for finite sums of sizes; but before we can evaluate

Countable Additivity, we need to make sense of infinite sums. One

way would be to define infinite sums of sizes directly in terms

of another primitive relation: aggregation of infinite collections of

regions. This would put Countable Additivity on par with Finite

24 Why think this? One reason is operationalism, the view that physical quantitieslike size must be defined in terms of measurement procedures, and operations likeaddition of sizes must likewise reduce to some mechanical procedure that can beperformed on the things that are measured. Another reason is nominalism: if size

properties arent reified, then their algebraic structure must latch directly onto thethings that bear the properties, as there isnt anything else for that structure todescribe. I dont find either of those reasons compelling, but even so I think theprinciple is true.


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Additivity as a defining condition for sums of sizes. But theres

another way of spelling out infinite sums of sizes, which is the stand-

ard way of making sense of infinite sums of numbers. This way lever-

ages the account we already have of finite sums. First we precisely

explicate the limit of a sequence of determinates.25 Then we defineFN:25

the value of an infinite sum of sizes to be the limit of the sequence

of finite sums. Using numbers to represent sizes and operating on

them with the

symbol tacitly calls on precisely this definition;

so it was implicit in the original statement of Countable Additivity.

This definition of an infinite sum of sizes makes no appeal at

all to aggregation of infinitely many things. Still, there is such an

infinite aggregation relationfor mereological sums, we have this

in Definition 4. This raises a substantive question: is the size of anaggregate of infinitely many things always equal to the infinite sum

of their sizes? That is, is the following true?

(22) If x is the aggregate of a countable set of regions S, then

the size of x is the sum of the sizes of the members of S.

(Countable Additivity)

This principle could go either way. Infinite sums of sizes are defined

apart from infinite aggregations of regions; there is no conceptual

reason why the two need coincide. Unlike Finite Additivity, Count-

able Additivity does not spring from the requirements for extensive

qualities: it might fail and yet leave size structure intact.

The present escape route involves denying Countable Additivity

in the particular case where aggregation is the disjoint fusionrelation. One could deny this by rejecting additivitythe limit of

a sequence of sizes might not reliably yield the size of an infinite

aggregateor by rejecting fusionsthe mereological sum might

not be the right relation to play the aggregation role for sizes. Ill

consider each of these options.

4.3 Non-standard mereology

Forrest levels his version of the argument against standard mere-

ology; in particular Weak Supplementation (2002). This move

25 We can define limit in terms of the ordering of sizes. Denote this order , andlet the interval [p, q] be the set of sizes {r : p r q}. Then we say limpk = p

iff forevery interval [q, r] that includes p (p = q = r, unless p is s bottom or top), thereexists some integer K such that for all k > K, pk [q, r].


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appears to miss the point: my strengthened argument does not

depend on any supplementation principle, so how could rejecting

Weak Supplementation constitute a response? Only if this denial is

followed by another: denying that fusions have countably additive

sizes. If Weak Supplementation fails, then the definition of fusion

(Definition 4) pulls apart in many ways from its usual theoreti-

cal roleand so in particular it becomes plausible that the fusion

relation so defined does not play the aggregation role for measure

theory.

When Weak Supplementation fails, some sets of regions have

more than one fusion; a region may fuse a set and yet fail to contain

any of its members; indeed, the fusion may be a proper part of some

member. These bizarre consequences suggest that in the absenceof Weak Supplementation, the standard definition of fusion fails

to capture a theoretically important natural relation, and fails to

capture the ordinary intuitive concept of composition.26 This is notFN:26

terribly surprising, seeing as Definition 4 was framed with standard

mereology in mind.

Lets distinguish fusion from another, closely related concept:

(D8) A region u is the least upper bound of S (for short: lubS)

iff: for every region x, x u iff x contains every y S.

(Antisymmetry guarantees uniqueness.) The least upper bound of S

is the smallest region that contains every element of S. If Remainder

Closure holds, then x is a fusion of S ifand only if x is the least upper

bound of S. So in standard mereological contexts, the two conceptshave the same extension. This fact, I submit, is the main motivation

for the standard definition of fusion. But when supplementation

principles fail, these two concepts pull apart from one another, and

so we lose the main reason for accepting the definition. In particular,

we lose our reason for thinking that the definition captures the right

relation to ground our measure theory.

Instead we should turn to the least upper bound. This relation

gets along without Weak Supplementation better than the fusion

relation: a least upper bound is unique and contains its members,

at least. This suggests that we should replace our original statement

of Countable Additivity with this one:

26 Cf. Forrest 2007. Kit Fine argues for a general conclusion along the same lines(2007).


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(23) For any countable set of pairwise-disjoint regions S,

m(lubS) =

xS m(x). (Lub Additivity)

This revision makes room for a consistent theory of topologically

gunky spaceat the price of adopting (in Forrests words) a semi-

standard mereology.

Forrests model for such space is the open sets with measure-

zero differences smudged (i.e. measure-zero equivalence classes

of open sets). The ontology of Forrests space is richer than Tarskis

but poorer than Arntzeniuss. We can see this by considering the

Cantor Regions from Section 3 once again. Arntzeniuss space

includes the Big Interval, the Cantor Regions, the Cantor Fusion

(the mereological sum of the Cantor Regions), and the Cantor Dust

(the difference between the interval and the Cantor Fusion). Tarskisspace, meanwhile, includes the Big Interval and the Cantor Regions,

but countenances neither the Cantor Fusion (as an entity distinct

from the Big Interval) nor the Cantor Dust. Forrests space includes

the Big Interval, the Cantor Regions, and the Cantor Fusion (better

called the Cantor Lub)but there is no dusty region making

up the difference between the Cantor Fusion and the interval. This

might strike some as the right balance, since the Cantor Dust is the

most bizarre of the regions under consideration. But accepting this

ontology sacrifices Weak Supplementation: without the Dust, the

interval has a proper part with nothing left over. If you paint each

of the Cantor Regions in Forrests space, you thereby succeed in

painting the Cantor Fusion but fail to paint the Big Intervalandtheres no way to finish the job without repeating some of your

work, since every part of the Big Interval has paint on it. This is an

unnecessarily drastic concession.

4.4 Finitely additive gunk

Returning to more hospitable mereology, why think that additivity

holds for countably infinite fusions? One might motivate this with

an infinite supertask, such as the Pudding Task.27 Say we have aFN:27

countable set of disjoint regions r1, r2, . . . which together compose

a finite region s, all in a world entirely free of pudding. At step

27 Forrest argues along these lines (1996). For a more elaborate example, seeHawthorne and Weatherson 2004.


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one, fill r1

with chocolate pudding; at step two fill r2

, and so on.

When all of the steps have been carried out, s is entirely filled with

chocolatey goodness. So the total amount of pudding in the world

is m(s). The total amount of pudding served is m(r1) +m(r2)+ . . ..

So if m(s) is greater than the sum, then there is more pudding in the

world than was served. Whence this extra pudding?

Intuition presses us toward this principle: the sizes of the finite

fusions should converge to the size of the infinite fusion. If we think

of the regions involved as also forming a convergent sequencethe

finite fusions converge to the infinite fusionthen theres anoth-

er way to say this: intuitively, the measure function should be

continuous.

Intuition favors continuous functions, yet continuity often fails inour dealings with the infinite.28 Consider, for instance, the PaintingFN:28

Task. Suppose we have an infinite sequence of blank pages labeled

with the natural numbers, and a large can of paint. At step one,

paint the first page; at step two paint the second page, and so on.

After any finite number of steps, there are infinitely many blank

pages. Yet when the supertask is complete, there are no blank

pages left at all. Again continuity fails: the limit of a sequence of

infinities is certainly not zero, so the limit gives the wrong result for

this supertask. Why shouldnt the Pudding Task be similar to the

Painting Task in this respect? As the number of steps carried out

approaches infinity, the amount of pudding approaches a particular

amountbut when the number of steps reaches infinity, there isa different amount. There is a break at infinity. Surprising, but

hardly impossible.

One might worry that completing such supertasks would make

it possible to violate certain physical principles: a one-mile bridge

could be built with only half a miles worth of concrete, by building

it up from the right set of parts. One available reply is that such

physical principles canbe violated: in worlds with the kind of space

in question, it may be physically possible to build a bridge on

the cheap in just this way. If the world has the right conservation

laws, though, then the supertask cannot be carried out. It is a little

puzzling that a law of physics might rule out an infinite sequence of

operations without rulingout any finite subsetof them, but this kind

28 This line of thought is influenced by conversation with Frank Arntzenius.


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of thing happens. For instance, in a Newtonian world, conservation

of momentum rules out an infinite sequence of collisions between

vanishingly small balls at vanishingly small distances from one

another, without ruling out any particular collision (Laraudogoitia

1996). The physical law is inconsistent with the initial conditions

that give rise to the infinite collision scenarioand so the law

rules the scenario out. Similarly, in worlds where conservation of

concrete is a law of nature, the bridge-building supertask cant be

carried out. Whatever initial conditions might give rise to it are

physically impossible.

One might also worry because measure supervenience fails. If the

size of a region is not determined by the sizes of a collection of

disjoint things that exhaustively compose it, doesnt that make itssize a sort of spooky, free-floating thing? One reply is tu quoque:

we are already accustomed to exactly this situation in pointy space,

since an uncountable collection of point-sized regions may be

rearranged to make up a region of any size whatsoever. If this

is acceptable, we might equally well learn to live without size

supervenience for countably infinite fusions of gunky regions. But

the spookiness worry might be raised more generally: what fixes

the size of a region, if not the sizes of its parts? For pointy space, the

answer to this question is written in the annals of classical measure

theory. For topologically gunky space it turns out to be a difficult

question.

The natural place to look for a model for finitely additive, topo-logically gunky space is Tarskis open regular sets. Tarskian regions

have the mereology and topology of their representative open sets;

it would be nice to similarly fix the size of a region to be the

(Lebesgue) measure of its representative. But this wont do.29 ToFN:29

see why not, consider once more the Cantor Regions, labeled a, b1,

b2, c1, c2, c3, c4, . . .. Recall: the size of a is14 , the combined sizes of

the bs is 18 , the combined sizes of the cs is1

16 , and so on; all of

these together add up to 12 ; and the fusion of the Cantor Regions

is the unit interval. Now let the Big Pieces be a, the cs, the es, etc.,

and let the Little Pieces be the bs, the ds, the fs, etc. Since the Big

Pieces are twice as big as the Little Pieces, and all together they

add up to1

2 , the Big Pieces sizes add up to1

3 and the Little Pieces

29 Thanks to John Hawthorne for forcing me to get clear on this.


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sizes add up to1

6 . Now consider Big Cantor, the fusion of the BigPieces, and Little Cantor, the fusion of the Little Pieces. How big are

they? According to the proposed rule, the measure of Big Cantor is13 and the measure of Little Cantor is

16 .

30 But note: the fusion of BigFN:30

Cantor and Little Cantor is the unit interval. So on this account of

size, Finite Additivity breaks down!

Fixing sizes based on the open regular sets was a bad idea. The

closed regular representatives do no better the closed representa-

tive of Big Cantor has size 56 , and the closed representative of Little

Cantor has size 23 ; these dont add up to 1 either.31 Appealing to

FN:31

a regions representative equivalence class is even less helpful: by

adding and subtracting regions like the Cantor Dust, we can find

members of each equivalence class with any given positive size.Which of these should we choose? The pointy representatives dont

pin down the sizes of Big and Little Cantor.

Heres another way of characterizing this problem: Tarskian

space includes regions that are not Jordan-measurable.32 To defineFN:32

Jordan measure, we start with a collection of elementary regions

whose sizes are already well understoodtypically rectangles (in

one dimension: intervals). We can calculate the measure of any

finite fusion of disjoint rectangles by addition. Call these finite

fusions the blocks. We then define the outer measure of a region x

as the lower limit of the sizes of the blocks that contain x, and the

inner measure as the upper limit of the sizes of the blocks that are

part of x. If the outer and inner measure agree, then x is Jordan-measurable, and their common value is the Jordan measure of x.

Jordan measure is finitely additive; in effect, Jordan measure starts

with the sizes of elementary regions and unrolls the consequences

of Finite Additivity for the rest. But there are regions whose sizes

30 Let I, U1 , U2, . . ., and V1, V2, . . . be the representative open sets for the unitinterval, the Big Pieces, and the Little Pieces, respectively, and let U =

Ui and

V =

Vi. U is an open regular set. This is because every point in I is either (i) amember of U, (ii) a member of V, or else (iii) a boundary point of both U andV. So the closure of U is IV, and the interior of clU is U. Thus U is BigCantors representative. Furthermore,since Lebesgue measure is countably additive,m(U) =

m(Ui) =

13 . Similar considerations hold for Little Cantor.

31 The closed representatives of Big Cantor and Little Cantor are the closures oftheir open representatives: namely I V and IU, respectively.

32 Jordan measure is to the Riemann integral as Lebesgue measure is to theLebesgue integral (Frink 1933).


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cant be determined in this wayand, sadly, Big Cantor and Little

Cantor are among them. Big Cantors inner measure is 13

, and its

outer measure is 56 . In pointy space, Lebesgue measure captures

these regions by going beyond finite collections of rectangles to

countable collectionsbut since Countable Additivity fails here,

this wont help.

One response to this failure is to cut the offending regions loose,

either by admitting that Tarskian space includes peculiar regions

that lack sizes, or else by restricting the space to include just the

Jordan-measurable regions. The latter option involves giving up

Unrestricted Composition for infinite collections of regions. These

are costs we hoped to escape in gunky space, but at any rate they

arent much worse than the costs faced by points: these two optionsparallel the two responses to non-measurable sets in Section 3.1.

For a model following the latter option, represent gunky regions

with Jordan-measurable open regular sets, and assign regions the same

parts, connection, and sizes as their representatives. The mereology

of this space is not quite standard: while every finite set of regions

has a fusion, some infinite sets do not: for instance, neither the Big

Pieces nor the Little Pieces have a fusion.33FN:33A second response to the failure of Jordan measure is to introduce

spooky sizes which are not pinned down by elementary regions

and additivity considerations. In fact, there are infinitely many

finitely additive measures that extend Jordan measure to all of the

Tarskian regions.34

We can assign Big Cantor any size between itsFN:34

33 The mereology satisfies the system CEM: it forms a Boolean algebra, but not acomplete Boolean algebra.

34 This is an application of a general result that Garrett Birkhoff (1967: 185 ff.) saysis due essentially to A. Tarski: if S is a subalgebra of a Boolean algebra A, then anyfinitely additive measure function defined on S has a finitely additive extension overall of A.

If a A and a S, we can extend the measure on S to the larger subalgebra ofelements of the form (b a) (c a) (where b a = b (b a)). We set the size ofsuch an element to be outerm,S(b a)+ innerm,S(c a), where outerm,S(x) = inf{m(s) :x s S} and innerm,S(x) = sup{m(s) : x s S}. Birkhoff shows (it isnt difficult)that this is finitely additive and agrees with m on S. The measure is extended to theentire algebra A using an induction principle (When A is uncountable this relies onthe Axiom of Choice).

Note that in this argument innerm,S(b a)+ outerm,S(c a) yieldsa measure justaswell, and any weighted average of measures is itself a measure. Therefore as long asthere is some element of A whose outer and inner measures disagree, m has infinitelymany finitely additive extensions.


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inner measure of1

3 and its outer measure of5

6 , and go on to fixsizes for the rest of the unsized regions in a consistent fashion.

All of the measure functions we obtain agree on simple cases like

rectangles, and all of them obey additivity. What could favor one

of these measure functions over any other? For instance, what

might determine that Big Cantor is truly bigger than Little Cantor,

rather than the other way around? The truth about the sizes of

such regions might be a matter of brute fact, but this introduces a

host of brute facts where on the classical story about measure we

have very few. There is, then, a model of Tarskian space with a

finitely additive measure function; in fact, there are infinitely many

non-isomorphic models. The problem is not that there is no way of

assigning sizes it is that there are too many.This leaves us with two kinds of finitely additive space. The

first kind requires giving up either Measurability or an infinite

composition principle, and the second kind requires giving up

a supervenience principle. These costs are considerable, but not

unthinkableremember, the point lover faces analogous compro-

mises. Besides these costs, merely finitely additive sizes lead to

counter-intuitive discontinuities, but intuitions are untrustworthy

when it comes to questions about continuity at infinity. Note finally:

uncountable additivity (insofar as uncountable sums make sense) is a

principle that pointy space has long since abandoned. Are countably

infinite sets more sacred? Finitely additive space is closer to the

Whiteheadian picture than either Arntzeniuss or Forrests alterna-tivesit is full-fledged gunk in the mereological, topological, and

measure-theoretic senses. In it we have a reasonable candidate for

at least some of the gunk theorists purposes.

5. conclusion

I have presented a strong argument against naive Whiteheadian

space: a small set of independently plausible conditions on gunky

space turns out to be inconsistent. The gunk theorist can respond

in several ways. The options on the table include Arntzeniuss

measure-theoretic gunk that violates the No Boundaries principle,Forrests space that introduces revisionary measure theory by way

of revisionary mereology, and a version of Tarskis geometry where


26/28



sizes disobey Countable Additivity. Each of these options goes

against certain intuitions: roughly, Arntzeniuss space sacrifices

an intuitive bit of topology, Forrests space an intuitive bit of

mereology, and Tarskis space an intuitive bit of measure theory.

ButI insist: theusualpointymodels of space violate intuitionas well,

as curiosities like the BanachTarski theorem (among many others)

go to show. We have to live with a bit of counter-intuitiveness in

any case.

How we choose among these theories will depend on what we

are choosing them for. If the goal is a theory of space that is

adequate for actual-world physics, Arntzeniuss measure-theoretic

gunk seems far and away the most satisfactory. If on the other hand

the main concern is capturing commonsense spatial reasoning, oneof the finitely additive models may be better. And if the question is

what spatial structures are broadly possible, then (short of further

argument to the contrary) possibilities abound.

One lesson to draw from this discussion is a reminder that

innocent-seeming premises can lead quickly into hidden incoheren-

cieswe should heed Humes admonition to be modest in our

pretensions.35 An account may be intuitive without being consis-FN:35

tent, especially when extrapolations to infinity are involved. But

there are babies as well as bath-water in intuitions tub: while the

naive conception of gunk is problematic, the core idea displays a

hardy resilience. In particular, my argument certainly does not rule

out gunky space as a physically interesting candidate for our actualuniverses space-time, as a metaphysical possibility, or as a consis-

tent framework for spatial reasoning. Avoiding contradictions does

force certain counter-intuitive results on usany theory of space

has its puzzles and surprisesbut we should expect that much.

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Queries in Chapter 11

Q1. Author correction is not clear.

jeff russell

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